A kinetic theory of dense fluids

A new kinetic equation is developed which incorporates the desirable features of the Enskog, the Rice-Allnatt, and the Prigogine-Nicolis-Misguich kinetic theories of dense fluids. Advantages of the present theory over the latter three theories are (1) it yields the correct local equilibrium hydrodynamic equations, (2) unlike the Rice-Allnatt theory, it determines the singlet and doublet distribution functions from the same equation, and (3) unlike the Prigogine-Nicolis-Misguich theory, it predicts the kinetic and kinetic-potential transport coefficients. The kinetic equation is solved by the Chapman-Enskog method and the coefficients of shear viscosity, bulk viscosity, thermal conductivity, and self-diffusion are obtained. The predicted bulk viscosity and thermal conductivity coefficients are singular at the critical point, while the shear viscosity and self-diffusion coefficients are not.     

Stationary extended kinetic theory for a gas of hard spheres

Summary Analytical and numerical results for the stationary, spatially homogeneous distribution function of a gas of ?hard-sphere? particles are reported. Both removal and scattering effects are accounted for. In the case of only removal, comparison is also made with results obtained in the frame of alternative models proposed for approximating the exact ?hard-sphere? model.     

Renormalized kinetic theory of nonequilibrium many-particle classical systems

The far-from-equilibrium statistical dynamics of classical particle systems is formulated in terms of self-consistently determined phase-space density response, fluctuation, and vertex functions. Collective and single-particle effects are treated on an equal footing. Two approximations are discussed, one of which reduces to the Vlasov equation direct interaction approximation of Orszag and Kraichnan when terms that are explicitly due to particles are removed.Work performed under the auspices of the U.S. Department of Energy.     

Nonlinear hyperbolicity in kinetic theory of a gas mixture

Summary A system of nonlinear hyperbolic conservation equations, arising in the study of an evolution problem of a mixture of gases of interacting particles in the presence of only removal effects, is illustrated. Explicit analytical solutions to such system are obtained and commented on both mathematical and physical grounds.

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Riassunto Nel presente lavoro si illustra un sistema di equazioni iperboliche non lineari di conservazione, che si incontra nello studio di un problema di evoluzione per una miscela di gas di particelle interagenti in presenza di soli effetti di rimozione. Soluzioni analitiche esplicite di tale sistema sono costruite e commentate su base sia matematica che fisica.

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Резюме Рассматривается система нелинейных гиперболических уравнений сошранения, возникающих при исследовании проблемы эволюции смеси газов взаимодействующих частиц в присутствии зффектов удаления. Получаются точные аналитические решения для такой системы. Проводится обсуждение полученных результатов с математической и физической точек зрения

     

Long-range correlations in kinetic theory

It is shown that macroscopic correlations in a fluid are conserved for macroscopically long times. The equations of conservation can be written in a form independent of the density of the fluid and are therefore valid for a liquid as well as for a gas. The possibility of developing a kinetic theory of turbulence on the basis of these equations (along the lines of V. N. Zhigulev and of S. Tsugé) is indicated.The contents of this paper formed part of the Ph.D. thesis submitted by the author under the supervision of Prof. Harold Grad to the Department of Mathematics, New York University and issued as NYU-Courant Institute of Mathematical Sciences Technical Report MF-72, October 1973.     

An analytical study of space-dependent evolution problems in particle transport theory

Summary Space-dependent evolution problems arising in particle transport theory are analytically studied via a systematic application of the Boltzmann equation. Some explicit solutions, that can improve our knowledge of the spatial effects in such a class of problems, are constructed and briefly commented on a physical ground.

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Riassunto Problemi spaziali di evoluzione che si incontrano nella teoria del trasporto di particelle sono studiati analiticamente per mezzo di una sistematica applicazione dell'equazione di Boltzmann. Alcune soluzioni esplicite, che possono migliorare la nostra conoscenza degli effetti spaziali in tale classe di problemi, sono costruite e commentate brevemente su base fisica.

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Резюме Аналитически, используя уравнение Больцмана, исследуются проблемы пространственной эволюции, зозниакющие в теории переноса частиц. Конструируются некоторые точные решения, которые могут уточнить наше понимание пространственных эффектов в таком классе проблем. Проводится обсуждение этих решений.

     

Dissociative attachment of an electron to a molecule: kinetic theory

Test particles interact with a medium by means of a bimolecular reversible chemical reaction. Two species are assumed to be much more numerous so that they are distributed according to fixed distributions: Maxwellians and Dirac's deltas. Equilibrium and its stability are investigated in the first case. For the second case, a system is constructed, in view of an approximate solution.     

Hamiltonian and Brownian systems with long-range interactions: IV. General kinetic equations from the quasilinear theory

We develop the kinetic theory of Hamiltonian systems with weak long-range interactions. Starting from the Klimontovich equation and using a quasilinear theory, we obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. This equation is valid at order 1/N in a proper thermodynamic limit and it coincides with the kinetic equation obtained from the BBGKY hierarchy. For N→+∞, it reduces to the Vlasov equation governing collisionless systems. We describe the process of phase mixing and violent relaxation leading to the formation of a quasistationary state (QSS) on the coarse-grained scale. We interpret the physical nature of the QSS in relation to Lynden-Bell’s statistical theory and discuss the problem of incomplete relaxation. In the second part of the paper, we consider the relaxation of a test particle in a thermal bath. We derive a Fokker-Planck equation by directly calculating the diffusion tensor and the friction force from the Klimontovich equation. We give general expressions of these quantities that are valid for possibly spatially inhomogeneous systems with long correlation time. We show that the diffusion and friction terms have a very similar structure given by a sort of generalized Kubo formula. We also obtain non-Markovian kinetic equations that can be relevant when the auto-correlation function of the force decreases slowly with time. An interesting factor in our approach is the development of a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems.     

Existence theory of the linear equations appearing in the Chapman-Enskog solutions to two kinetic equations for liquids

The linear operators appearing in the Chapman-Enskog solutions to Kirkwood's Fokker-Planck kinetic equation and to Rice and Allnatt's kinetic equation are studied in this article. Existence proofs are given for the linearized Chapman-Enskog equations involving either the Fokker-Planck or the Rice-Allnatt operators. It is shown that the Fokker-Planck and Rice-Allnatt operators, defined in the domain appropriate to kinetic theory, are essentially self-adjoint. It is also shown that the spectrum of either of these operators coincides with the spectrum of the self-adjoint extension of the corresponding operator.Sloan Foundation Fellow 1968–70. Guggenheim Fellow 1969–70.     

On the kinetic theory of a dense gas of rough spheres

The revised Enskog equation for a dense gas of rough spheres is considered. TheH theorem and the conservation equations are discussed.     

A kinetic theory and benchmark predictions for polymer-dispersed, semi-flexible macromolecular rods or platelets

The goals of this paper are: to present a mean-field kinetic theory for the hydrodynamics of macromolecular high aspect ratio rods or platelets dispersed in a polymeric solvent; and, to apply the formalism to predict the impact due to a polymeric versus viscous solvent on the classical Onsager isotropic-nematic equilibrium phase diagram and on the monodomain response to imposed steady shear. The kinetic theory coupling between the nanoscale rods or platelets and the polymeric solvent is incorporated through a mean-field potential that reflects the enormous particle-polymer surface area and the particle-polymer interactions across this interfacial area. To determine predictions of this theory on the equilibrium and sheared monodomain phase diagrams, we present a reduction procedure which approximates the coupled Smoluchowski equations for the polymer chain probability distribution function (PDF) and the nano-particle orientational PDF in favor of a coupled system of equations for the rank 2 second-moment tensors for each PDF. The reduced model consists of an 11-dimensional dynamical system, which we solve using continuation software (AUTO) to predict the modified Onsager equilibrium phase diagram and the modified Doi-Hess shear phase diagram due to the physics of polymer-particle surface interactions.     

Hamiltonian and Brownian systems with long-range interactions: V. Stochastic kinetic equations and theory of fluctuations

We developed a theory of fluctuations for Brownian systems with weak long-range interactions. For these systems, there exists a critical point separating a homogeneous phase from an inhomogeneous phase. Starting from the stochastic Smoluchowski equation governing the evolution of the fluctuating density field of Brownian particles, we determine the expression of the correlation function of the density fluctuations around a spatially homogeneous equilibrium distribution. In the stable regime, we find that the temporal correlation function of the Fourier components of density fluctuations decays exponentially rapidly, with the same rate as the one characterizing the damping of a perturbation governed by the deterministic mean field Smoluchowski equation (without noise). On the other hand, the amplitude of the spatial correlation function in Fourier space diverges at the critical point T=T_c (or at the instability threshold k=k_m) implying that the mean field approximation breaks down close to the critical point, and that the phase transition from the homogeneous phase to the inhomogeneous phase occurs sooner. By contrast, the correlations of the velocity fluctuations remain finite at the critical point (or at the instability threshold). We give explicit examples for the Brownian Mean Field (BMF) model and for Brownian particles interacting via the gravitational potential and via the attractive Yukawa potential. We also introduce a stochastic model of chemotaxis for bacterial populations generalizing the deterministic mean field Keller-Segel model by taking into account fluctuations and memory effects.     

A kinetic model for step flow growth in molecular beam epitaxy

A terrace-step-kink model for epitaxial step flow growth of steps with no bonds along them is derived from kinetic arguments. The model is combined with an existing model for the steps that have strong bonding along them to describe steps of arbitrary orientation in terms of densities of adatoms, step adatoms and kinks. A planar steady-state solution for a simplified version of the model is constructed and analyzed. Different mass transport mechanisms are modeled that result in different far-from-equilibrium behavior, confirming that edge diffusion is the main factor stabilizing the steps during growth. Furthermore kinetic Wulff shapes are constructed from the calculated step velocities.     

A variational principle for boundary value problems in kinetic theory

A variational principle which applies directly to the integrodifferential form of the linearized Boltzmann equation is introduced. Extremely general boundary conditions and collision terms are allowed. For a class of interesting problems, the value of the functional to be varied is shown to be closely related to quantities of great physical interest. The formalism is applied to the treatment of plane Couette flow for different forms of the collision term (BGK model, rigid spheres, Maxwell's molecules).Research sponsored by the Air Force Office of Scientific Research under contract F 61(052)-68-C-0020, through the European Office of Aerospace Research, OAR, United States Air Force.     

A reinterpretation of dense gas kinetic theory

In dense gas kinetic theory it is standard to express all reduced distribution functions as functionals of the singlet distribution function. Since the singlet distribution function includes aspects of correlated particles as well as describing the properties of freely moving particles, it is here argued that these aspects should more clearly be distinguished and that it is the distribution function for free particles that is the prime object in terms of which dense gas kinetic theory should be expressed. The standard equations of dense gas kinetic theory are rewritten from this point of view and the advantages of doing so are discussed.     

Multi-velocity and multi-temperature model of the mixture of polyatomic gases issuing from kinetic theory

In this paper, we consider Euler-like balance laws for mixture components that involve macroscopic velocities and temperatures for each different species. These laws are not conservation laws due to mutual interaction between species. In particular, source terms that describe the rate of change of momentum and energy of the constituents appear. These source terms are computed with the help of kinetic theory for mixtures of polyatomic gases. Moreover, if we restrict the attention to processes which occur in the neighborhood of the average velocity and temperature of the mixture, the phenomenological coefficients of extended thermodynamics can be determined from the computed source terms.     

Quantum kinetic theory of confined electrons in a strong time dependent field

A gauge-invariant Green’s function approach to the quantum transport of spatially confined electrons in strong electromagnetic fields is presented. The theory includes mean field and exchange effects, as well as collisions and initial correlations. It allows for a self-consistent treatment of spectral properties and collective effects (plasmons), on one hand, and nonlinear field phenomena, such as harmonic generation and multiphoton absorption, on the other. It is equally applicable to electrons in quantum dots, ultracold ions in traps and valence electrons of metal clusters.     

On the invariance of constitutive equations according to the kinetic theory of gases

Iterative techniques for solving the Boltzmann equation in the kinetic theory of gases yield expressions for the stress tensor and heat flux vector that are analogous to constitutive equations in continuum mechanics. However, these expressions are not generally invariant under the Euclidean group of transformations, whereas constitutive equations in continuum mechanics are usually required to be by the principle of material frame indifference. This disparity in invariance properties has led some previous investigators to argue that Euclidean invariance should be discarded as a contraint on constitutive equations. It is proven mathematically in this paper that the results of the Chapman-Enskog iterative procedure have no direct bearing on this issue. In order to settle this question, it is necessary to examine mathematically the effect of superimposed rigid body rotations on solutions of the Boltzmann equation. A preliminary investigation along these lines is presented which suggests that the kinetic theory is consistent with material frame indifference in at least a strong approximate sense provided that the disparity in the time scales of the microscopic and macroscopic motions is extremely large—a condition which is usually a prerequisite for the existence of constitutive equations.On leave from Stevens Institute of Technology, Hoboken, New Jersey 07030.